N2008 P1 Q11

The equations of three planes \(p_1,p_2,p_3\) are \(2x-5y+3z=3\), \(3x+2y-5z=-5\), \(5x+\lambda y+17z=\mu\), respectively, where \(\lambda\) and \(\mu\) are constants. When \(\lambda=-20.9\) and \(\mu=16.6\), find the coordinates of the point at which these planes meet.[2]

The planes \(p_1\) and \(p_2\) intersect in a line \(l\).

1. Find a vector equation of \(l\).[4]
2. Given that all three planes meet in the line \(l\), find \(\lambda\) and \(\mu\).[3]
3. Given instead that the three planes have no point in common, what can be said about the values of \(\lambda\) and \(\mu\)?[2]
4. Find the cartesian equation of the plane which contains \(l\) and the point \((1,-1,3)\).[4]